Question: Simplify and expand the following expression: $ \dfrac{3p}{p - 6}+\dfrac{p}{3p + 7} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(p - 6)(3p + 7)$ Multiply the first term by $\dfrac{3p + 7}{3p + 7}$ $ \begin{align*} \dfrac{3p}{p - 6} \times \dfrac{3p + 7}{3p + 7} & = \dfrac{(3p)(3p + 7)}{(p - 6)(3p + 7)} \\ & = \dfrac{9p^2 + 21p}{(p - 6)(3p + 7)}\end{align*} $ Multiply the second term by $\dfrac{p - 6}{p - 6}$ $ \begin{align*} \dfrac{p}{3p + 7} \times \dfrac{p - 6}{p - 6} & = \dfrac{(p)(p - 6)}{(3p + 7)(p - 6)} \\ & = \dfrac{p^2 - 6p}{(3p + 7)(p - 6)}\end{align*} $ Now we have: $ = \dfrac{9p^2 + 21p}{(p - 6)(3p + 7)} + \dfrac{p^2 - 6p}{(3p + 7)(p - 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{9p^2 + 21p + p^2 - 6p}{(p - 6)(3p + 7)} $ $ = \dfrac{10p^2 + 15p}{(p - 6)(3p + 7)}$ Expand the denominator: $ = \dfrac{10p^2 + 15p}{3p^2 - 11p - 42}$